One mole of an ideal gas at \(25^{\circ} \mathrm{C}\) expands in volume from \(1.0 \mathrm{~L}\) to \(4.0 \mathrm{~L}\) at constant . temperature. What work (in \(J\) ) is done if the gas expands against vacuum \(\left(P_{\text {cxtemal }}=0\right)\) ? (a) \(-4.0 \times 10^{2}\) (b) \(-3.0 \times 10^{2}\) (c) \(-1.0 \times 10^{2}\) (d) Zero

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. In particular, it examines the transfer of energy to and from objects, especially in the form of heat, and how this affects the properties of those objects. One of the most important concepts in thermodynamics is the idea of work, which in a physical context, is the energy transferred to or from an object by means of a force acting on it over a distance.

When dealing with gases, thermodynamics often focuses on processes where a gas expands or contracts and how this affects not only the internal energy of the gas but also the work done by or on the gas. The laws of thermodynamics help us understand and calculate these energy changes, especially when considering real-world applications like engines and refrigerators.

Understanding Work in Gas Expansion

In the context of gas expansion, work is done by the gas when it expands against an external pressure. However, when this expansion is against a vacuum, the external pressure is zero, and therefore no work is performed according to thermodynamic principles. It's vital to recognize that work is defined as the force exerted over a distance. In a vacuum, with no opposing force, there is no work to be done by the expanding gas.

The ideal gas law is a foundational equation in thermodynamics and physical chemistry, expressed as PV=nRT. Here’s what each letter represents:

• P stands for pressure,
• V is the volume,
• n is the number of moles,
• R is the ideal gas constant, and
• T is the temperature in Kelvin.

The ideal gas law provides a relation between these four quantities and is used to calculate one if the other three are known. It’s an idealized equation based on the assumption that the particles in a gas move independently and have no interactions with each other, apart from perfectly elastic collisions.

Even though this is an idealization, the law offers good approximations for most gases under standard conditions. When applying the ideal gas law to calculate work done by a gas during an expansion or contraction, it's essential to consider the external conditions, such as pressure, that the gas is subjected to. In the case of expansion against a vacuum, even though the volume changes, the absence of external pressure (as denoted by zero in the equation) means no work is done.

Gas expansion against a vacuum is a specific scenario in thermodynamics where a gas expands without any external pressure resisting its expansion. Under normal circumstances when a gas expands, it does so against atmospheric pressure or the pressure exerted by another gas or container. In these cases, the gas does work, as it must exert energy to overcome the external pressure.

However, a vacuum presents no such opposition, which fundamentally changes the dynamics of expansion. Since there's nothing resisting the gas’s expansion, the concept of 'work done by the gas' becomes redundant.

No Work Performed

In the exercise provided, an ideal gas expands from 1.0 L to 4.0 L without external pressure. Mathematically, the work done by the gas can be calculated using the area under the pressure-volume (P-V) curve in a P-V diagram. However, since the external pressure is zero throughout the expansion, the curve would lie along the volume axis, and the area – indicative of work done – also amounts to zero, reinforcing the physical concept that no work is done by the gas.